Question

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are in the plane which is perpendicular to the plane containing $\overrightarrow{c}$ and $\overrightarrow{d}$ then $(\overrightarrow{a}\times \overrightarrow{b})\times (\overrightarrow{c}\times \overrightarrow{d})$ is

• A. $0$
• B. $1$
• C. $\overrightarrow{a}.\overrightarrow{d}+\overrightarrow{b}.\overrightarrow{c}$
• D. $noneofthese$

Answer 1

The correct option is A $0$

$(\overrightarrow{a}\times \overrightarrow{b})$ is a plane $\perp$ to both $\overrightarrow{a}$ and $\overrightarrow{b}$

$(\overrightarrow{c}\times \overrightarrow{d})$ is a plane $\perp$ to both $\overrightarrow{c}$ and $\overrightarrow{d}$

Now $\overrightarrow{a}$ and $\overrightarrow{b}$ are in plane which is perpendicular to plane $\overrightarrow{c}$ and $\overrightarrow{d}$

$\therefore$ $\overrightarrow{a}\times \overrightarrow{b}$ is in with zero degree with $\overrightarrow{c}\times \overrightarrow{d}$

$\therefore$ $(\overrightarrow{a}\times \overrightarrow{b})\times (\overrightarrow{c}\times \overrightarrow{d})=0$